Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2007, Article ID 83863, 14 pages doi:10.1155/2007/83863 Research Article Performance Analysis of Blind Subspace-Based Signature Estimation Algorithms for DS-CDMA Systems with Unknown Correlated Noise Keyvan Zarifi and Alex B Gershman Department of Communication Systems, Darmstadt University of Technology, Merckstraße 25, 64283 Darmstadt, Germany Received October 2005; Revised 30 March 2006; Accepted April 2006 Recommended by Vincent Poor We analyze the performance of two popular blind subspace-based signature waveform estimation techniques proposed by Wang and Poor and Buzzi and Poor for direct-sequence code division multiple-access (DS-CDMA) systems with unknown correlated noise Using the first-order perturbation theory, analytical expressions for the mean-square error (MSE) of these algorithms are derived We also obtain simple high SNR approximations of the MSE expressions which explicitly clarify how the performance of these techniques depends on the environmental parameters and how it is related to that of the conventional techniques that are based on the standard white noise assumption Numerical examples further verify the consistency of the obtained analytical results with simulation results Copyright © 2007 K Zarifi and A B Gershman This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited INTRODUCTION In recent years, extensive research efforts have been devoted to develop different strategies for multiuser detection in DSCDMA systems [1] One of the most challenging problems in multiuser detection is that of the effect of unknown multipath channel which may result in a significant mismatch between the actual user signature and its presumed value used in the multiuser detection algorithms Since signature mismatches may cause substantial degradation in the symbol detection performance [2–5], considerable attention has been paid to designing accurate signature estimation techniques at the receiver These techniques may be classified into the training-based [6, 7] and blind [3, 4, 8–13] methods In the training-based approaches, each user transmits a sequence of pilot symbols which is known at the receiver where the user signature is estimated by computing the correlation between the received data and this sequence In nonstationary environments, a reliable signature estimate requires periodic transmission of the pilot sequence This may cause a considerable reduction of the bandwidth efficiency [2, 3] and has been a strong motivation to develop alternative blind estimation approaches which not require transmission of the pilot sequence A promising trend among this type of methods is the subspace-based techniques [3, 8–11] The latter techniques exploit the facts that the user signals occupy a lowdimensional subspace in the observation space, and that the signature of each particular user belongs to a subspace defined by its associated spreading code A typical assumption used in these techniques is that the additive ambient noise is temporally white, and, hence, the signal subspace can be extracted using eigendecomposition of the received data covariance matrix However, in practice this assumption may be violated [14, 15] It is well known that in the presence of correlated noise, the signal subspace cannot be identified from the subspace spanned by the eigenvectors associated with the largest eigenvalues of the data covariance matrix Therefore, some alternative approaches should be employed to identify the signal subspace in the correlated noise case One of such approaches has been proposed by Wang and Poor [15] Their technique is based on the assumption that the receiver contains two well-separated antennas so that the receiver noise is spatially white Using this fact, the signal subspace can be obtained from the crosscorrelation between the received antenna data Hereafter, we refer to this technique by Wang and Poor as the WP algorithm 2 Using a single antenna at the receiver, another technique that addresses the problem of correlated noise has been proposed by Buzzi and Poor [16] It is based on the assumption that the noise is a circular Gaussian process while the transmitted symbols are noncircular BPSK signals In such a case, it has been shown in [16] that the signal subspace can be directly identified using the singular value decomposition of the data pseudocovariance matrix Hereafter, we refer to the technique by Buzzi and Poor as the BP algorithm Although the performance of the conventional (white noise assumption-based) signature waveform estimation techniques has been well studied in the literature [9, 10, 17– 19], only a little effort has been made to analyze the performance of the estimation algorithms proposed for the unknown correlated noise case In this paper (see also [20]), we use the first-order perturbation theory to derive approximate expressions for the MSE of the channel vector estimates obtained by the WP and BP algorithms Under several mild assumptions, simple high SNR approximations of these MSE expressions are also obtained The derived MSE expressions clarify how the performance of the algorithms depends on the parameters such as the number of data samples, the received power of the user of interest, and the noise covariance matrix The effect of the spreading factor and the channel length on the performance of the algorithms is also studied It is shown that the performance of the algorithms depends not only on SNR but also on the direction of the eigenvectors of the noise covariance matrix To clarify this fact, we fix the eigenvalues of the noise covariance matrix and find the sets of eigenvectors which maximize (minimize) the MSEs of the channel vector estimates Moreover, over all noise covariance matrices with fixed trace, we obtain those which correspond to the extremal values of the MSEs It is shown that both the maximum and the minimum values of the MSEs are obtained when the noise covariance matrix is rank deficient As the trace of the noise covariance matrix is equal to the average noise power, the latter observation shows that the performance of the algorithms may be more sensitive to a low-rank interference than to a full-rank noise with the same average power We also show that in the presence of white noise, the performances of the WP and BP algorithms are identical to that of the conventional Liu and Xu (LX) algorithm [9] that was developed for the white noise case Assuming that the SNR is high and the WP algorithm is used to estimate the channel vector between the user of interest and the first antenna, it is proved that the estimation performance is independent from the noise covariance matrix and the user received power at the second antenna We use the latter property to show that when the receiver is equipped with multiple antennas, the second antenna can be arbitrarily chosen at high SNRs The rest of this paper is organized as follows In Section 2, we introduce the signal model A brief overview of the LX, WP, and BP algorithms is provided in Section Section presents our main theoretical results on the performance of the WP and BP algorithms Simulation results validating our EURASIP Journal on Advances in Signal Processing analysis are presented in Section Conclusions are drawn in Section SIGNAL MODEL Consider a K-user synchronous DS-CDMA system.1 The received continuous-time baseband signal can be modelled as [3] x(t) = ∞ K m=−∞ k=1 Ak bk (m)wk t − mTs + v(t), (1) where Ts is the symbol period, v(t) is the zero-mean additive random noise process, and Ak , bk (m), and wk (t) denote the received signal amplitude, the mth data symbol, and the signature waveform of the kth user, respectively Note that bk (m) can be drawn from a complex constellation, and, hence, in the general case x(t) is complex valued Throughout the paper, we use the following common assumptions (A1) The chip sequence period is equal to the symbol period, that is, the short spreading code is considered [22] (A2) The user channels are quasistatic, that is, the corresponding impulse channel responses not change during the whole observation period [9] (A3) The duration of the channel impulse response of each user is much shorter than the symbol period Ts , so that the effect of intersymbol-interference (ISI) can be neglected [9, 22] (A4) The transmitted symbols and noise are zero-mean random variables Moreover, transmitted symbols of each user are unit-variance i.i.d variables, independent from those of the other users, and also independent from the noise [9] Note that (A1) is common for many multiuser techniques proposed for DS-CDMA systems as most of these algorithms require the received signal x(t) to be cyclostationary This, in turn, necessitates the use of short spreading codes [23] Let Lc be the spreading factor and let ck = [ck [1], ck [2], , ck [Lc ]]T denote the discrete spreading sequence associated with the kth user where (·)T stands for the transpose and ck [i] can be either real or complex valued According to assumptions (A1) and (A2), the signature waveform of this user can be expressed as [9] Lc ck [l]hk t − lTc , wk (t) = (2) l=1 where hk (t) is the channel impulse response of the kth user and Tc = Ts /Lc is the chip period The synchronous case is mainly considered for the sake of notational brevity It is straightforward to extend our analysis to the asynchronous [15] as well as the multiple-antenna [21] DS-CDMA systems K Zarifi and A B Gershman Let us assume that hk (t) is zero outside the interval [0, αTc ], where L − ≤ α < L and L is a positive integer Lc Sampling (1) From assumption (A3), it follows that L in the interval corresponding to the nth transmitted symbol of each user and ignoring the first L − samples that are contaminated by ISI, the ISI-free received sampled data vector can be written as [9] the signal subspace eigenvalues Due to the fact that the noise is white, range(W) = range(Us ), or, equivalently, Us and Un span the signal and noise subspaces, respectively Without any loss of generality, we assume that h1 is the channel vector of interest As any column of Un is orthogonal to all vectors in range(Us ), we have [9] UH w1 = T1 h1 = 0, n K Ak bk (n)wk + v(n), x(n) = (3) k=1 where x(n) = [x(nTs + LTc ), x(nTs + (L + 1)Tc ), , x(nTs + Lc Tc )]T , wk = [wk (LTc ), wk ((L + 1)Tc ), , wk (Lc Tc )]T , and v(n) = [v(nTs + LTc ), v(nTs + (L + 1)Tc ), , v(nTs + Lc Tc )]T Note that the similar data model also holds when the effects of chip waveform at the transmitter and chip matched filtering at the receiver are taken into account [21] Using (2), we have that the signature vector wk can be written as ⎡ ⎤ ck [L] · · · ck [1] ⎢c [L + 1] · · · ck [2] k ⎢ wk = ⎢ ⎢ ⎣ ck L c · · · ck Lc − L + ⎥ ⎥ ⎥ hk ⎥ ⎦ Ck hk , (4) where hk = [hk (0), hk (Tc ), , hk ((L − 1)Tc )] As the spreading code of the user of interest is known at the receiver, if the channel vector hk is estimated, then wk can be obtained from (4) Hence, throughout this paper we consider the problem of channel vector estimation rather than that of the signature vector estimation For the sake of consistency, we also assume without any loss of generality that hk is a unit Euclidean norm vector ( hk = 1) [9], that is, the normalization factor is absorbed in Ak One can present (3) in a more compact form as [9] x(n) = Wb(n) + v(n), (5) where W = [A1 w1 , A2 w2 , , AK wK ], b(n) = [b1 (n), b2 (n), , bK (n)]T BLIND CHANNEL ESTIMATION 3.1 The LX algorithm R E x(n)x(n) H = WW H + σv I, (6) where σv I is the noise covariance matrix, I is the identity ma2 trix, and σv = E{|v(t)|2 } is the noise variance The matrix (6) can be eigendecomposed as R = Us Un where T1 UH C1 is an Lc − L+1 − K × L matrix From (8), it n follows that T1 is not full rank Assuming that rank(T1 ) = L− 1, the null space of T1 is spanned by h1 , and, therefore, up to an arbitrary phase rotation, h1 can be uniquely determined as a nontrivial solution to (8) subject to h1 = Note also that if C1 is a full-rank matrix, then rank(T1 ) = L − is equivalent to [9] dim range C1 ∩ range(W) = 1, Ωs + σv I σv I UH s , UH n (7) where Us consists of the eigenvectors associated with the K largest eigenvalues which are the diagonal elements of Ωs + σv I, and Ωs is a diagonal matrix whose diagonal elements are (9) where dim{·} stands for the dimension of a subspace Equation (9) is the necessary and sufficient condition of signature identifiability using the LX algorithm [9] In practical scenarios, the data covariance matrix R is not known exactly and can be estimated as R= N N x(n)x(n)H (10) n=1 As a result, Un is estimated as Un that consists of the eigenvectors associated with the smallest Lc − L+1 − K eigenvalues of R Substituting Un in lieu of Un in (8) and solving the obtained equation in the least square (LS) sense, we have that the estimated channel vector h1 is given by [9] h = M C H Un UH C , n (11) where M{·} stands for the normalized eigenvector associated with the smallest (minor) eigenvalue Using the firstorder perturbation theory, the mean-square of the encountered estimation error δh1 = h1 − h1 can be approximately written as [18] E The LX algorithm assumes that the noise is white In such a case, from assumption (A4) and (5) we have [9] (8) δh1 ≈ σv † T N H F w1 − − Us Ωs UH + σv Us Ωs UH w1 , s s (12) where T† is the pseudoinverse of T1 and · F stands for the Frobenius norm of a matrix Assuming that the signatures of different users are orthogonal to each other, that is, wiH w j = wi δ i j , (13) where δi j stands for the Kronecker delta, the MSE expression (12) can be significantly simplified Note that due to multipath effects, the orthogonality assumption of the signature vectors does not perfectly hold in practice However, CDMA codes are deliberately designed so that even after passing through a frequency selective channel, the cross correlations between different user signatures are as small as possible 4 EURASIP Journal on Advances in Signal Processing Hence, in most practical scenarios, (13) is an acceptable assumption [1] It directly follows from (13) that w w w1 , , , K w1 w2 wK Us = , (14) 2 Ωs = diag A2 w1 , A2 w2 , , A2 wK K where the right-hand side of (19) is the singular value decomposition (SVD) of R(12) It is clear that range(U(1) ) = s range(W(1) ) and range(U(2) ) = range(W(2) ) For the sake of s simplicity but without any loss of generality, let us consider only the channel vector between the first user and the first antenna Then, we have [15] H (1) U(1) w1 = T(1) h(1) = 0, 1 n Substituting (14) into (12) and using (13) yields E δh1 ≈ † 2 σv T1 NA2 F 1+ σv w1 A2 H (15) A2 w1 , the MSE of If SNR is high enough, that is, σv the channel estimate is further simplified to δh1 E 2 σv T† ≈ NA2 F (20) (16) where T(1) U(1) C1 is an Lc − L + − K × L matrix n (1) If rank(T1 ) = L − 1, then up to an arbitrary phase rotation, h(1) is the unique nontrivial solution to (20) subject to h(1) = [15] In practice, R(12) can be estimated as R(12) = N N H x(1) (n)x(2) (n) (21) n=1 which results in the following estimate of h(1) [15] Equation (16) can be considered as a reasonable approximation of (12) in the high SNR regime Note that an expression equivalent to (16) has been derived for the MSE of the estimated signature, C1 h1 , in [9] where U(1) consists of the left singular vectors associated with n the Lc − L + − K smallest singular values of R(12) 3.2 WP algorithm 3.3 It is well known that if the white noise assumption does not hold, then the signal subspace is not identical to the subspace spanned by the eigenvectors associated with the K largest eigenvalues of R and, consequently, the LX algorithm cannot be directly applied to obtain a reliable estimate of h1 To deal with this problem, the WP algorithm assumes that the receiver is equipped with two well-separated antennas such that the noise is spatially uncorrelated between them Similar to (5), the sampled received data vectors are given by Another approach to solve the problem of channel estimation in presence of unknown correlated noise has been proposed in [16] Without requiring the second antenna, this algorithm is based on the assumption that the transmitted symbols are drawn from the BPSK constellation (bk (n) = ±1) and the noise is a circular Gaussian process It directly follows from the latter assumption that i = 1, 2, x(i) (n) = W(i) b(n) + v(i) (n), (17) (i) (i) where i is the antenna index, W(i) = [A(i) w1 , A(i) w2 , , (i) A(i) wK ], v(i) (n) is noise at the ith antenna, and A(i) and K k (i) wk = Ck h(i) are the received amplitude and the signature k vector of the kth user at the ith antenna, respectively The covariance matrix corresponding to the sampled received data vector at each antenna is given by [15] R(i) H H E x(i) (n)x(i) (n) = W(i) W(i) + Σ(i) , v i = 1, 2, (18) H where Σ(i) = E{v(i) (n)v(i) (n)} As the noise is uncorrelated v between the antennas, we have [15] R(12) H E x(1) (n)x(2) (n) = W(1) W(2) ⎡ (1) = Us U(1) n H H ⎤ (2) Ω(12) ⎢Us ⎥ s ⎦, ⎣ 0 U(2) H n (19) H h(1) = M CH U(1) U(1) C1 , 1 n n (22) BP algorithm E v(n)v(n)T = (23) E{x(n)xT (n)} be the pseudocovariance matrix of Let R the sampled received data Using (5) along with (23), we have [16] R = WWT = Us Un Ωs 0 VH s , VH n (24) where Ωs is a diagonal matrix whose diagonal elements are the nonzero singular values of R and the columns of Us are the corresponding left singular vectors It is easy to show that range(Us ) = range(W) [16], and, hence, UH w1 = T1 h1 = 0, n (25) where T1 UH C1 It can be observed that T1 is an Lc − L + n − K × L matrix and the unique identification of h1 from (25) requires that rank(T1 ) = L − [16] In practice, similar to the LX and WP algorithms, h1 can be estimated by H h1 = M CH Un Un C1 , (26) K Zarifi and A B Gershman where Un is the matrix containing the left singular vectors associated with the Lc − L + − K least singular values of R, and R= N N x(n)x(n)T (27) n=1 Note that both the MSE expressions (28) and (32) depend on Σ(1) only through tr Σ(1) Ψ To study the paramv v eters which have impact on the value of tr Σ(1) Ψ , we first v should note that if the channel vector is uniquely identifiable, then rank(T(1) ) = rank(Ψ) = L − Moreover, we have τ is the sample estimate of R where null(·) stands for the null-space of a matrix.2 The effects of different parameters on the value of tr Σ(1) Ψ are v separately clarified in the following discussion PERFORMANCE ANALYSIS 4.1 WP algorithm In order to evaluate the performance of the WP algorithm, we use the first-order perturbation theory to prove the following theorem Theorem Assume that h(1) is estimated using (22) Then, the first-order perturbation theory-based approximation of the MSE of the estimation error δh(1) = h(1) − h(1) is given by 1 E δh(1) ≈ †H † (1) (1) H tr Σ(1) Ψ w1 R(12) R(2) R(12) w1 , v N (28) where tr(·) stands for the trace of a matrix and U(1) T(1) n Ψ †H † H T(1) U(1) n (29) Moreover, if the following conditions hold: H (i) (i) wk wl(i) = wk δkl , i = 1, 2, (30) (31) (2) A(2) w1 λmax Σ(2) v , then (28) reduces to E δh(1) ≈ tr Σ(1) Ψ v NA(1) , Proof See Appendix A Note that the average received power of the first user at the second antenna is equal to the right-hand side of (31), while the average noise power at the same antenna is lower bounded by the left-hand side because v(2) (n) = tr Σ(2) ≥ λmax Σ(2) v v Effects of Lc and L As Σ(1) and Ψ are positive (semi-) definite matrices, it follows v that tr(Σ(1) Ψ) is real and nonnegative Note that the projecv tion of Σ(1) onto null(Ψ) does not have any effect on the value v of tr(Σ(1) Ψ) which depends only on the projection of Σ(1) v v onto range(Ψ) Therefore, the larger the projection of Σ(1) v onto null(Ψ), the smaller the value of tr(Σ(1) Ψ) Using the v latter fact, the effect of the spreading factor and the channel length on tr(Σ(1) Ψ), and, consequently, on the performance v of the WP algorithm can be explained as follows From (34) it can be observed that if either the spreading factor Lc increases or the channel length L decreases, then dim{null(Ψ)} increases In the latter case, the projection of the columns of Σ(1) onto null(Ψ) becomes larger, and, therefore, their conv tribution to the value of tr(Σ(1) Ψ) becomes smaller v Effect of the eigenvectors of Σ(1) v The directions of the eigenvectors of Σ(1) with respect to the v eigenvectors of Ψ have a considerable impact on the value of tr(Σ(1) Ψ) To show this, let us eigendecompose Ψ as v Ψ = ΠΘΠH , (32) where λmax (·) stands for the maximum eigenvalue E dim null(Ψ) = Lc − L + − rank(Ψ) = Lc − 2(L − 1), (34) where Π = [π π · · · π L−1 ] is an Lc − L+1 × L − matrix whose columns are the orthonormal eigenvectors associated with the decreasingly-ordered positive eigenvalues of Ψ that are the diagonal elements of Θ = diag{θ1 , θ2 , , θL−1 } In contrary to rank(Ψ), m rank(Σ(1) ) may not be known In v fact, rank(Σ(1) ) may vary from m = for the case of coherent v interference to m = Lc − L + for the case of full-rank noise Let us consider an arbitrary value of m and eigendecompose Σ(1) as v (33) Hence, if SNR at the second antenna is reasonably high, it is guaranteed that (31) holds Using this observation along with the fact that (30) approximately holds in most practical scenarios, we can view (32) as a simple approximation of (28) in the high SNR regime It explicitly clarifies the MSE of the estimated channel vector in terms of the environmental parameters such as the received power of the user of interest at the first antenna, the number of data samples as well as the noise covariance matrix Σ(1) v (35) Σ(1) = Uv Γv UH , v v (36) where Uv is an Lc − L + × m matrix whose orthonormal columns are the eigenvectors associated with the decreasingly-ordered positive eigenvalues of Σ(1) which are v the diagonal elements of Γv = diag{γ1 , γ2 , , γm } It should be noticed from (29) that range(W(1) ) is a K-dimensional subspace in null(Ψ) 6 EURASIP Journal on Advances in Signal Processing The value of tr(Σ(1) Ψ), and, hence, the MSE expresv sions (28) and (32) critically depend on the direction of the columns of Uv relative to the columns of Π To explain this fact, let us fix the matrix Γv and find the matrices Uv max and Uv which maximize and minimize tr Σ(1) Ψ , respectively v It can be shown [24, 25] that τ1 max tr Σ(1) Ψ v Uv γi θi , = τ1 = min{L − 1, m}, (37) i=1 and Uv max is given by ⎧ ⎪ π1 π2 · · · πm , ⎨ Uv max = ⎪ ⎩ Π Π⊥ −L+1 , m if m ≤ L − 1, if m > L − 1, (38) where Π⊥ is an Lc − L + × l matrix whose l ≤ τ columns l are arbitrarily chosen from a set of τ orthonormal vectors in null(Ψ) According to (38), for a fixed Γv , the MSE expressions (28) and (32) are maximal if the first τ1 columns of Uv and Π coincide In turn, we have [24, 25] tr Uv Σ(1) Ψ v = ⎧ ⎪0, ⎪ ⎪ ⎨ m−τ ⎪ ⎪ ⎪ ⎩ i=1 if m ≤ τ, γτ+i θL−i , if m > τ, (39) and Uv is given by ⎧ ⎪Π⊥ , ⎨ m if m ≤ τ, ⎩ Π⊥ π L−1 · · · π L−(m−τ) , τ if m > τ Uv = ⎪ (40) According to (40), the necessary condition to minimize the MSE expressions (28) and (32) is that the first τ2 min{m, τ } columns of Uv are in null(Ψ) Note that the matrix Σv = Uv Γv UH has the maximum projection vmin onto null(Ψ), that is, the space spanned by the eigenvectors associated with the τ2 largest eigenvalues of Σv is in null(Ψ) Assuming that the average noise power at the first antenna is given by eo , that is, E v(1) (n) m = tr Σ(1) = v γ i = eo , (41) i=1 we can also obtain the extremal values of the MSE expressions (28) and (32) as follows Since for any pair of positive (semi-) definite matrices Σ(1) and Ψ we have [25] v tr Σ(1) Ψ ≤ λmax (Ψ)tr Σ(1) , v v (42) it directly follows that Moreover, it is obvious that among all noise covariance matrices with m γi = eo , those in the form of i= Σ(1) = Π⊥ Γv Π⊥ m m v tr ≤ θ1 eo , (43) where, assuming that the largest eigenvalue of Ψ is unique, (43) holds with equality if and only if Σ(1) = eo π π H v (44) (45) result in the MSE expressions (28) and (32) equal to zero It is interesting to observe from (44) and (45) that, given the average noise power at the first antenna, both the maximal and the minimal values of the MSE of the channel vector estimate are obtained when the noise covariance matrix is rank deficient As a rank deficient covariance matrix can be attributed to a narrow-band interference, it follows that the performance of the WP algorithm can be more sensitive to a narrow-band interference than a full-rank colored noise Now, let us consider two important particular scenarios in which the WP algorithm may be used and discuss the pertaining results White noise scenario: if the noise at the first antenna is (1) white, that is, Σ(1) = σv I, then (32) reduces to v E δh(1) ≈ (1) σv T(1) NA(1) † F (46) which is equal to the derived MSE of the LX algorithm in (16) Hence, even though the WP algorithm is proposed to estimate the channel vector in the presence of unknown correlated noise, it is also applicable to the white noise scenario In the latter case, the performance of the WP algorithm is identical to that of the LX algorithm Multiple antenna systems: it follows from (32) that if the SNR at the second antenna is high enough so that (31) holds, then the MSE of the channel vector estimate between the user of interest and the first antenna is independent of Σ(2) and v the received power of this user at the second antenna Let us consider a receiver with M > antennas which are spatially separated so that the noises between the first antenna and all the other antennas are uncorrelated Moreover, assume that the SNR is high enough: λmax Σ(i) v (i) A(i) w1 , i = 2, , M, (47) and that we aim to estimate the channel vector between the first user and the first antenna using the WP algorithm Since this algorithm is based on processing of the data crosscorrelation matrix between the first antenna and another well-separated auxiliary antenna, we have to choose the auxiliary antenna among the M − available antennas However, it directly follows from (32) that if the aforementioned assumptions hold, the performance of the channel vector estimate is insensitive to the choice of such an antenna, that is, the auxiliary antenna can be chosen arbitrarily 4.2 Σ(1) Ψ v H BP algorithm The following theorem quantizes the performance of the BP algorithm Theorem Assume that the channel vector is estimated using the BP algorithm Then, the first-order perturbation theorybased approximation of the MSE of the estimation error K Zarifi and A B Gershman δh1 = h1 − h1 is given by E δh1 ≈ where the matrix Uv which maximizes tr(Σv Ψ) is H †H tr Σv Ψ RT + Σv ΨΣv w R N T R † w1 , (48) where Σv = E v(n)v(n)H , H R = WW + Σv , (50) H Un T† T† UH 1 n Ψ (49) (51) Moreover, if (13) holds and λmax Σv A2 w1 , δh1 ≈ tr Σv Ψ NA2 (53) As can be observed from (53), in the high SNR regime the MSE of the channel vector estimate of the BP algorithm can be expressed in terms of the noise covariance matrix, power of the received signal, and the number of data samples Note that if the channel vector is uniquely identifiable from the BP algorithm, we have rank(Ψ) = L − Comparing (53) with (32), it can be readily shown that the effect of the spreading factor and the channel length on both the WP and BP algorithms are similar Moreover, following a discussion similar to that of Section 4.1, we can obtain the extremal values of tr(Σv Ψ), and, consequently, those of the MSE expression (53) Let us first eigendecompose Ψ as H tr Σv Ψ Uv τ1 γi θi , i=1 τ1 = min{L − 1, q}, (57) ⎪ ⎪ ⎪ ⎩ if q ≤ τ, q−τ i=1 γτ+i θL−i , if q > τ, if q ≤ τ, (56) if q > τ Πτ π L−1 · · · π L−(q−τ) , (58) (59) Comparing (56)–(59) with (37)–(40), it can be observed that the conclusions which follow (37)–(40) can be easily extended to the BP algorithm, and, hence, we not repeat them for the sake of brevity Let us also consider the case that the average noise power q is given by eo , that is, tr(Σv ) = i=1 γi = eo In such a case, assuming that the largest eigenvalue of Ψ is unique, the noise covariance matrix which maximizes tr Σv Ψ is given by H Σv = eo π π (60) Moreover, over all noise covariance matrices Σv with q i=1 γi = eo , the value of tr(Σv Ψ) and, consequently, that of the MSE expression (53) is zero if and only if ⊥H Σv = Πq Γv Πq (61) Similar to the WP algorithm, it follows from (60) and (61) that the performance of the BP algorithm can be more sensitive to the narrow-band interference than to the full-rank noise If noise is white, that is, Σv = σv I, the MSE expression (53) reduces to (55) (i) for any given Γv , Uv = ⎧ ⎪0, ⎪ ⎪ ⎨ ⊥ where Uv contains the orthonormal eigenvectors associated with the positive eigenvalues of Σv which are ordered decreasingly as the diagonal elements of Γv = ⊥ diag{γ1 , γ2 , , γq } Denoting Πl as an Lc − L + × l matrix whose columns are orthonormal vectors in null(Ψ), we have = if q > L − 1; , ⊥ E max tr Σv Ψ if q ≤ L − 1, ⎧ ⊥ ⎪Π , ⎨ q (54) where Π = [π π · · · π L−1 ] contains the orthonormal eigenvectors associated with the positive eigenvalues of Ψ and Θ = diag{θ1 , θ2 , , θL−1 } is the diagonal matrix that contains the decreasingly-ordered positive eigenvalues Let us denote q rank(Σv ) and eigendecompose Σv as Σv = Uv Γv UH , v Π ⊥ Πq−L+1 (ii) for any given Γv , Uvmin = ⎪ ⎩ Proof See Appendix B Ψ = ΠΘΠ , Uv max = ⎪ ⎩ where the matrix Uv which minimizes tr(Σv Ψ) is (52) then (48) reduces to E ⎧ ⎪ π1 π2 · · · πq , ⎨ δh1 ≈ σv T† NA2 F (62) Hence, the performances of the BP and the LX algorithms are identical in the white noise scenario Therefore, the BP algorithm can also be applied to estimate the channel vector in the white noise case without any estimation performance loss as compared to the conventional LX algorithm Another interesting relationship between the WP and BP algorithms follows from comparing (32) and (53) Let the users transmit BPSK modulated symbols and let the receiver be equipped with two well-separated antennas such that noise is spatially uncorrelated between them Also, let EURASIP Journal on Advances in Signal Processing 10−1 10−2 10−2 MSE 100 10−1 MSE 100 10−3 10−3 10−4 10−4 10−5 10−5 10−6 −15 −10 −5 10 15 20 25 10−6 −15 30 −10 −5 SNR (dB) 10 SNR (dB) SNR(2) = −20 dB SNR(2) = −10 dB SNR(2) = dB Experimental Analytical: (28) Analytical: (32) Figure 1: The MSE of the estimated channel versus SNR The WP algorithm 15 20 25 30 SNR(2) = 10 dB SNR(2) = 20 dB SNR(2) = 30 dB Figure 3: The MSE of the estimated channel versus SNR at the first antenna for different values of SNR at the second antenna The WP algorithm 10−2 100 10−1 10−3 MSE MSE 10−2 10−4 10−3 10−4 10−5 10−5 50 100 150 200 250 10−6 −15 300 N −5 10 15 20 25 30 SNR (dB) Experimental Analytical: (28) Analytical: (32) Experimental Analytical: (48) Analytical: (53) Figure 2: The MSE of the estimated channel versus number of data samples The WP algorithm (30) and (31) hold and λmax Σ(1) v −10 Figure 4: The MSE of the estimated channel versus SNR The BP algorithm (1) A(1) w1 (63) Then, the MSE expressions (32) and (53) can be readily verified to coincide in the following two cases: when h(1) is es1 timated using the WP algorithm with both antennas, and when h(1) is estimated using the BP algorithm with only the first antenna SIMULATIONS In this section, we validate our analytical results via computer simulations In all the examples, we consider K = synchronous CDMA users that transmit BPSK-modulated symbols Each point of the simulation curves is the result of averaging over 200 Monte-Carlo realizations of the noise and transmission data sequences In Figures 1–8, Gold codes of length Lc = 31 are employed as the user spreading sequences K Zarifi and A B Gershman 10−2 102 100 10−2 MSE MSE 10−3 10−4 10−4 10−6 10−8 10−10 10−5 50 100 150 200 250 10−12 −15 300 −10 −5 N 10 15 20 25 30 SNR (dB) Σv Σv Σv Σv Σv Σv Σv Experimental Analytical: (48) Analytical: (53) Figure 5: The MSE of the estimated channel versus number of data samples The BP algorithm drawn randomly, q = drawn randomly, q = drawn randomly, q = 15 drawn according to (60) drawn according to (61), q = drawn according to (61), q = drawn according to (61), q = 15 Figure 7: MSEs of the estimated channel versus SNR for eo = 28 and different matrices Σv The BP algorithm 102 100 100 MSE 10−2 10−1 10−4 10−2 MSE 10−6 10−8 10−10 10−12 −15 10−3 10−4 −10 −5 10 15 20 25 30 10−5 SNR (dB) Uv drawn randomly Uv drawn randomly Uv drawn randomly Uv drawn according to (57) Uv drawn according to (59) Figure 6: The MSE of the estimated channel versus SNR for Γv = diag{20, 5, 3} and different matrices Uv The BP algorithm and channel vectors of length L = are independently drawn from a zero-mean white complex Gaussian process and then are scaled to become unit-norm vectors The ambiguity in the phase of the channel vector estimate is resolved by assuming that the phase of the first tap of the channel vector is known at the receiver In Figures 1–5 and 9, the received noise at each antenna is considered to be a circular Gaussian process such that [Σv ]i j , the (i, j)th entry of its covariance matrix, is equal to 0.7|i− j | In the figures where the MSE of 10−6 −15 −10 −5 10 15 20 25 30 SNR (dB) LX algorithm WP algorithm BP algorithm Figure 8: MSEs of the estimated channel versus SNR in the white noise environment The LX, WP, and BP algorithms the channel estimate is drawn versus SNR, it is assumed that N = 80 data samples are used to estimate the channel Figures 1–3 illustrate the accuracy of our analytical results derived for the WP algorithm In Figure 1, it is assumed that SNRs of all users at both antennas are identical and h(1) is estimated according to (22) The solid curve represents the 10 EURASIP Journal on Advances in Signal Processing 10−2 MSE 10−3 10−4 10−5 10 Channel length Experimental, Lc = 40 Analytical: (48), Lc = 40 Analytical: (53), Lc = 40 12 14 Experimental, Lc = 80 Analytical: (48), Lc = 80 Analytical: (53), Lc = 80 Figure 9: MSEs of the estimated channel versus L for Lc = 40 and Lc = 80 The BP algorithm MSE resulting from this estimate This curve is compared with our analytical results given by (28) and (32) It can be observed that both theoretical curves follow the experimental MSE curve with a good precision Note that when the SNR is very low, the channel vector estimation error is quite large and, hence, it could not be reliably predicted using the firstorder perturbation theory In such a condition, the analytical MSE curves obtained from (28) and (32) show a considerable discrepancy with the experimental MSE curve Figure depicts the experimental and the analytical MSE curves versus the number of data samples N In this figure, it is assumed that the received signal power from each user at each of the two antennas is equal to 10 dB Due to the fact that SNR is reasonably high, the theoretical curve (28) and its high SNR approximation (32) are almost indistinguishable from each other and they follow the experimental MSE curve with a good accuracy It can be observed from Figure that, when the number of data samples N is small, the small perturbation assumption is violated, and, hence, the accuracy of the analytical MSE curves decreases Figure shows the MSE of the estimated channel h(1) ver1 sus SNR at the first antenna (SNR(1) ) for different values of SNR at the second antenna (SNR(2) ) As expected from Section 4.1, the performance of the channel estimation is almost independent from the exact value of SNR(2) , unless SNR(2) is very low Figures 4–7 and show the performance of the BP algorithm and compare it to our analytical results In Figure 4, the experimental MSE curve is plotted versus SNR and is compared with the theoretical curves obtained from (48) and (53) As can be observed from the figure, the two theoretical MSE curves are very close to each other and also closely follow the experimental MSE curve for the SNRs higher than dB Figure shows the experimental and the theoretical curves drawn versus the number of data samples N for SNR equal to 10 dB As the figure shows, the theoretical curve (48) is precisely followed by its high SNR approximation (53) and both of them are very close to the experimental MSE curve Figure shows the experimental MSE curves versus SNR for noise covariance matrices with identical Γv = diag{20, 5, 3} and different matrices of eigenvectors Uv Three random realizations of Uv as well as Uvmax and Uvmin are drawn and then using (55) the corresponding noise covariance matrices are obtained The BP algorithm is used to estimate the channel vector in the presence of a correlated noise with the so-obtained noise covariance matrices Figure confirms our theoretical results in Section 4.2 which state that the worst and the best MSE performances are obtained when Uv = Uvmax and Uv = Uvmin , respectively Note that if Uv = Uvmin , then, unlike the MSE expression (53), the experimental MSE performance is not equal to zero It is due to the fact that the MSE expression (53) is obtained using the first-order perturbation theory and even in the high SNR regime this expression has a slight difference with the experimental MSE Figure plots the experimental MSE curves versus SNR for noises with identical average energy of eo = Lc − L+1 = 28 and different covariance matrices For each value of q = 1, 5, and 15, one noise covariance matrix is drawn randomly and another one is obtained according to (61) A rank-one noise covariance matrix is also derived according to (60) Then, the BP algorithm is used to estimate the channel vector in the presence of correlated noise with the so-obtained noise covariance matrices Our analytical results in Section 4.2 are validated by observing that the worst and the best MSE performances are obtained when the noise covariance matrix follows (60) and (61), respectively In Figure 8, the performances of the LX, WP, and BP algorithms are tested in the white noise environment As predicted by our analysis in Section 4, all three methods have a nearly identical performance Figure shows the experimental and the theoretical MSE curves of the BP algorithm versus the channel length L for two different values of the spreading factors Lc = 40 and Lc = 80 In this example, we use random spreading codes rather than the optimized Gold codes The entries of these codes are randomly drawn from the set {−1, +1} From Figure we see that, as predicted in Section 4, the estimation performance decreases with increasing L When Lc = 80, the MSE of the channel vector estimate is significantly lower than that for Lc = 40 It can be observed that the curves corresponding to (48) and (53) are quite close to each other and, therefore, the use of the random spreading codes instead of the Gold codes retains the accuracy of (53) CONCLUSIONS In this paper, analytical expressions for the MSE of the signature waveform estimation techniques of [15, 16] have been K Zarifi and A B Gershman 11 derived Assuming that different user signature vectors are orthogonal, the simplified versions of these expressions have been also obtained for the high SNR regime The effect of the correlated noise on the performance of both algorithms has been studied It has been shown that the direction of the eigenvectors of the noise covariance matrix has a significant effect on the performance of both algorithms In particular, assuming that the eigenvalues of the noise covariance matrix are fixed, the noise covariance matrix eigenvectors corresponding to the extremal values of the MSEs have been obtained Over all noise covariance matrices with identical average noise power, the extremal values of the MSEs have been derived and it has been shown that both the maximal and the minimal values of the MSEs are achieved when the noise covariance matrix is rank deficient Moreover, it has been shown that at high SNRs and in the presence of white noise, the performance of these two techniques is identical to that of the conventional white noise-based technique of [9] In the high SNR regime, it has been proved that the performance of the technique proposed in [15] is independent from the noise covariance matrix and the user received power at the second auxiliary antenna This property has been generalized to the multiple antenna systems and it has been shown that for such systems the choice of the auxiliary antenna is arbitrary at high SNRs δh(1) E (1) H ≈ w1 R(12) †H H † (1) E δR(12) ΨδR(12) R(12) w1 (A.9) Let us introduce H Ξ E δR(12) ΨδR(12) (A.10) From (A.2) and (A.3) it follows that H Ξ = E R(12) ΨR(12) (A.11) Using (17) and (21) in (A.11) yields Ξ= N2 N N W(2) b( j) + v(2) ( j) E j =1 k=1 H H × b( j)H W(1) + v(1) ( j) Ψ × W (1) (A.12) (1) b(k) + v (k) H H × b(k)H W(2) + v(2) (k) Using (A.2) to simplify the resulting expression, we obtain Φ1 + Φ2 , N Ξ= (A.13) where APPENDICES A Inserting (A.5) into (A.8) and applying the expectation operation to the squared norm of the resulting expression, we have PROOF OF THEOREM Φ2 Since U(1) spans the null-space of R(12) , we have n H U(1) W(1) n = ΨW =W ( j)Ψv ( j)v (1) (2) H H Φ1 = E v(1) ( j)Ψv(1) ( j) W(2) W(2) (1) H H Ψ = = E tr v(1) ( j)Ψv(1) ( j) (A.2) δR(12) R(12) − R(12) , (A.3) δU(1) n U(1) − U(1) n n (A.4) Using the perturbation theory, the first-order approximation of δU(1) can be written as [9, 26, 27] n ≈ −R (1) H (12) † H δR (12) H U(1) , n (A.5) where −1 † ( j) , (A.14) (A.1) H = tr E v(1) ( j)v(1) ( j)Ψ To prove (28), we introduce δU(1) n E v ( j)v (2) H We also have Equations (A.1) and (29) yield (1) H E W(2) b( j)v(1) ( j)Ψv(1) ( j)bH ( j)W(2) Φ1 H R(12) = U(2) Ω(12) U(1) s s s = tr Σ(1) Ψ W(2) W v Φ2 = E v W(2) W(2) W(2) W(2) H H (A.15) , H ( j)Ψv ( j) E v(2) ( j)v(2) ( j) (1) = tr Σ(1) Ψ Σ(2) v v Substituting (A.15) into (A.13) and using (18), we obtain (A.16) tr Σ(1) Ψ R(2) v N Using (A.16) in (A.9) directly yields (28) To prove (32), first we use (19) and (30) to obtain Ξ= (A.6) Since (1) H (2) H H U(1) = s (1) w(1) w1 , , K (1) (1) w1 wK , Ω(12) s (1)H Un C1 h(1) ≈ 0, (A.7) it follows that the first-order estimate of δh(1) is given by (1) † δh(1) ≈ −T1 H (1) δU(1) w1 n (A.8) (1) (2) = diag A1 A1 (1) w1 U(2) = s (2) (1) w1 , , A(1) A(2) wK K K (2) w(2) w1 , , K (2) (2) w1 wK (2) wK , (A.17) 12 EURASIP Journal on Advances in Signal Processing Let where ˘ (1) w1 R (12) † (1) w1 (12) −1 = U(2) Ωs s H (1) U(1) w1 s w(2) (1) (2) (2) A1 A1 w1 (A.19) δh(1) ≈ Substituting R(2) from (18) into (A.20) and using (30) to simplify the result, we obtain E ≈ NA(1) 1+ 2 (2) A(2) w1 (A.21) (2) As for any w1 and Σ(2) , v H (2) (2) (2) w1 Σ(2) w1 ≤ w1 λmax Σ(2) , v v Lc −L+1 Lc −L+1 [Φ2 ]kl = g =1 = E [v]m [v∗ ]k E [v]l [v]∗ g Σv lg + Σv mg Σv ΨW = WH Ψ = Φ2 kl = g =1 H H ≈ w1 R† E δ RH Ψδ R R† w1 , (B.3) where R − R Σv = Σv ΨΣv (B.2) lk ΨΣv gk + ΨΣv + tr ΨΣv Σv Φ2 = Σv ΨΣv T + tr ΨΣv ΣT v Us = w w1 , , K w1 wK (B.5) H Ξ = E R ΨR (B.6) Expanding the right-hand side of (B.6) according to (27), and then using (B.2) to simplify the resulting expression, we obtain Ξ= Φ1 + Φ2 , N (B.7) lk (B.13) (B.14) , Ωs = diag A2 w1 , , A2 wK K Substituting (B.4) into (B.5), and then using (B.2) to simplify the result, we have Σv lk E δ RH Ψδ R gg Substituting (B.10) and (B.14) into (B.7) and using the resulting expression in (B.3), we obtain (48) To prove (53), we note that (13) along with (24) yield (B.4) Let us denote lg From (B.13) it directly follows that Using the procedure similar to that in Appendix A, it can be readily shown that Ξ lk Lc −L+1 (B.1) From (51) along with (B.1) it follows that δR mk Substituting (B.12) into (B.11), we obtain UH W = n (B.12) + E [v]m [v]∗ E [v]l [v]∗ g k = Σv According to (24), we have δh1 [Ψ]gm E [v]∗ [v]∗ [v]m [v]l , (B.11) g k E [v]∗ [v]∗ [v]m [v]l g k (A.22) PROOF OF THEOREM E m=1 where [·]k is the kth entry of a vector and the time index i has been dropped from v(i) for the sake of simplicity Since v is a multivariate circular Gaussian random vector, we have [28] then, when (31) holds, (32) directly follows from (A.21) This completes the proof B (B.10) where the second line of (B.10) can be derived using the same steps as in (A.15) To obtain Φ2 , it can be easily shown from (B.9) that H (2) (2) w1 Σ(2) w1 v (B.8) (B.9) = tr Σv Ψ W∗ WT , H tr Σ(1) Ψ (2) v w(2) R(2) w1 (1) (2) (2) NA1 A1 w1 tr Σ(1) Ψ v E v (i)v (i)Ψv(i)v (i) , Φ1 = E vH (i)Ψv(i) W∗ WT , (A.20) δh(1) T and (·)∗ stands for the conjugate Since the transmitted symbols are drawn from the BPSK constellation, we have Using (A.19) along with (28) yields E H ∗ Φ2 Substituting (A.17) into (A.18) and using (30), we have ˘ (1) w1 = E vH (i)Ψv(i)W∗ b(i)bT (i)WT , Φ1 (A.18) Vs = ∗ ∗ w w1 , , K w1 wK , (B.15) Let us denote ˘ w1 −1 R† w1 = Vs Ωs UH w1 s (B.16) Substituting (B.15) into the right-hand side of (B.16) and using (13), we have ˘ w1 = ∗ w1 A2 w1 (B.17) K Zarifi and A B Gershman 13 Using (B.17) in (48) results in the following expression for E{ δh1 }: δh1 E ≈ tr ΨΣv α1 + α2 + α3 , N w1 A4 (B.18) where T w1 w∗ W∗ WT , w1 w1 α1 α2 T ∗ w1 T w1 Σv , w1 w1 T w1 w1 tr ΨΣv α3 Σv ΨΣv (B.19) T ∗ w1 w1 It directly follows from (13) that α1 = A2 w1 (B.20) Noting that both Σv and Ψ are positive (semi-) definite matrices, it is easy to find an upper-bound for α2 and α3 as α2 = H w1 w Σv w1 w1 α3 = tr ΨΣv ∗ ≤ λ∗ Σv = λmax Σv , max H w1 w1 Σv ΨΣv w1 w1 ≤ λ∗ Σv ΨΣv max tr ΨΣv = λmax Σv ΨΣv tr ΨΣv ≤ ∗ λmax ΨΣv λmax Σv ≤ λmax Σv tr ΨΣv (B.21) Hence, if (52) holds, both α2 and α3 are negligible comparing to α1 Substituting (B.20) into (B.18) directly yields (53) This completes the proof ACKNOWLEDGMENTS This work was supported by the Wolfgang Paul Award Program of the Alexander von Humboldt Foundation and German Ministry of Education and Research REFERENCES ´ [1] S Verdu, Multiuser Detection, Cambridge University Press, Cambridge, UK, 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Zarifi received his B.S and M.S degrees both in electrical engineering from University of Tehran, Tehran, Iran, in 1997 and 2000, respectively From January 2002 until March 2005, he was with the Department of Communication Systems, University of Duisburg-Essen, Duisburg, Germany From April 2005, he has been with the Department of Communication Systems, Darmstadt University of Technology, Darmstadt, Germany, where he currently pursues his Ph.D From September 2002 until March 2003, he was a Visiting Scholar at the Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada His research interests include statistical signal processing, multiuser detection, blind estimation techniques, and MIMO communications Alex B Gershman received his Diploma and Ph.D degrees in radiophysics and electronics from the Nizhny Novgorod State University, Russia, in 1984 and 1990, respectively From 1984 to 1999, he held several full-time and visiting research appointments in Russia, Switzerland, and Germany In 1999, he joined the Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada, where he became a Full Professor in 2002 From April 2005, he has been with the Darmstadt University of Technology, Darmstadt, Germany, as a Professor and Head of the Communication Systems Group His research interests span the areas of signal processing and communications with primary emphasis on statistical signal and array processing, adaptive beamforming, MIMO and space-time communications, multiuser and multicarrier communications, radar and sonar signal processing, and estimation and detection theory He received the 2004 IEEE Signal Processing Society Best Paper Award He is a Fellow of the IEEE and a recipient of the 2001 Wolfgang Paul Award from the Alexander von Humboldt EURASIP Journal on Advances in Signal Processing Foundation, Germany, the 2002 Young Explorers Prize from the Canadian Institute for Advanced Research (CIAR), and the 2000 Premier’s Research Excellence Award, Ontario, Canada He is currently Editor-in-Chief of the IEEE Signal Processing Letters He was Associate Editor of the IEEE Transactions on Signal Processing (1999–2006) He is on Editorial Boards of EURASIP Journal on Wireless Communications and Networking and EURASIP Signal Processing Journal He is Vice-Chair of the Sensor Array and Multichannel (SAM) Technical Committee (TC) of the IEEE Signal Processing Society ... Zarifi and A B Gershman, ? ?Performance analysis of subspace-based signature waveform estimation algorithms for DS-CDMA systems with unknown correlated noise,” in Proceedings of 6th IEEE Workshop on... user signature vectors are orthogonal, the simplified versions of these expressions have been also obtained for the high SNR regime The effect of the correlated noise on the performance of both algorithms. .. performance of the algorithms may be more sensitive to a low-rank interference than to a full-rank noise with the same average power We also show that in the presence of white noise, the performances